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A006566 Dodecahedral numbers: n(3n-1)(3n-2)/2.
(Formerly M5089)
21
0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711, 45760, 52394, 59640, 67525, 76076, 85320, 95284, 105995, 117480, 129766, 142880, 156849, 171700, 187460, 204156 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Schlaefli symbol for this polyhedron: {5,3}

A093485 = first differences; A124388 = second differences; third differences = 27. - Reinhard Zumkeller, Oct 30 2006

One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). [From Daniel Forgues, May 14 2010]

From Peter Bala, Sep 09 2013: (Start)

a(n) = binomial(3*n,3). Two related sequences are binomial(3*n+1,3) (A228887) and binomial(3*n+2,3) (A228888). The o.g.f.'s for these three sequences are rational functions whose numerator polynomials are obtained from the fourth row [1, 4, 10, 16, 19, 16, 10, 4, 1] of the triangle of trinomial coefficients A027907 by taking every third term:

sum(n >= 1, binomial(3*n,3)*x^n )   = (x + 16*x^2 + 10*x^3)/(1-x)^4

sum(n >= 1, binomial(3*n+1,3)*x^n ) = (4*x + 19*x^2 + 4*x^3)/(1-x)^4

sum(n >= 1, binomial(3*n+2,3)*x^n ) = (10*x + 16*x^2 + x^3)/(1-x)^4.

(End)

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

Tanya Khovanova, Recursive Sequences

Hyun Kwang Kim, On Regular Polytope Numbers

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

FORMULA

G.f.: x(1+16x+10x^2)/(1-x)^4. a(n) = A000292(3n-3) = A054776(n)/6 = n*A060544(n).

a(n) = C(n+2,3) + 16 C(n+1,3) + 10 C(n,3)

a(0)=0, a(1)=1, a(2)=20, a(3)=84, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jul 24 2013

a(n) = binomial(3*n,3). a(-n) = - A228888(n). sum(n >= 1, 1/a(n) ) = 1/2*( sqrt(3)*Pi - 3*log(3) ). sum(n >= 1, (-1)^n/a(n) ) = 1/3*sqrt(3)*Pi - 4*log(2). - Peter Bala, Sep 09 2013

MAPLE

A006566:=(1+16*z+10*z**2)/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

Table[n(3n-1)(3n-2)/2, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 20, 84}, 40] (* Harvey P. Dale, Jul 24 2013 *)

PROG

(PARI) a(n)=n*(3*n-1)*(3*n-2)/2

(Haskell)

a006566 n = n * (3 * n - 1) * (3 * n - 2) `div` 2

a006566_list = scanl (+) 0 a093485_list  -- Reinhard Zumkeller, Jun 16 2013

CROSSREFS

A027907, A228887, A228888.

Sequence in context: A044207 A044588 A172221 * A205312 A211158 A154077

Adjacent sequences:  A006563 A006564 A006565 * A006567 A006568 A006569

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Henry Bottomley, Nov 23 2001

STATUS

approved

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Last modified August 28 07:22 EDT 2014. Contains 246162 sequences.